Evolutionary Wavelet Neural Network for Large Scale Function Estimation in Optimization

Transcription

1 AIAA Paper AIAA , th AIAA/ISSMO Multdscplnary Analyss and Optmzaton Conference, Portsmouth, VA, September 6-8, 006. Evolutonary Wavelet Neural Network for Large Scale Functon Estmaton n Optmzaton Debass Sahoo and George S. Dulkravch Department of Mechancal and Materals Engneerng Multdscplnary Analyss, Inverse Desgn, and Robust Optmzaton Center (MAIDROC Florda Internatonal Unversty, 0555 W. Flagler St., Mam, Florda 7, U.S.A. e-mal: dulkrav@fu.edu Web page: Ths paper descrbes a new method for constructng wavelet neural network n order to mprove the accuracy of predcton for mult-dmensonal functon spaces. An algorthm s developed usng the concept of evolutonary search n wavelet neural network. It helps n decreasng the computatonal effort needed for buldng the wavelet neural network. Several modfcatons to wavelet neural network are also suggested for mprovng ts performance n predctng non-lnear functon spaces. These algorthms were tested usng dverse test functons. These networks can be effectvely used as non-lnear system estmators for large scale optmzaton problems. I. Introducton Functon estmaton nvolves fndng the underlyng relatonshp from a gven set of nput-output dataset. Ths has been popular n varous applcatons such as pattern classfcaton, data mnng, system dentfcaton, and response surface constructon n optmzaton,,. The system dentfcaton problem s to estmate the underlyng system characterstcs usng a set of emprcal nput-output data. Recently, feed-foreword neural networks such as multlayer perceptron (MLP and radal bass functon networks (RBFN have been wdely used as an alternatve approach to functon approxmaton. They provde a generc black box functon representaton and have been shown to be capable of approxmatng any contnuous functon defned on a compact set of data wth arbtrary accuracy. Usng the concept of locally supported bass functon, such as RBFN, a class of wavelet neural network (WNN has become popular lately -7. It also uses the concept of wavelet decomposton. Combnng wavelet and neural networks dmnshes the weakness of each other 7. The resultng network s capable of handlng problems of moderately large dmenson and can be effcently constructed. Due to the radal structure of the wavelets, t can be consdered as a more general form of RBFN. Few modfcatons to such networks were suggested to deal wth outlners n tranng data 8. Ths helps n tranng the WNN wth nosy dataset. Constructng WNN nvolves estmatng the parameters n the nodes of the network and the weghts. Ths s done by mnmzng a cost functon whch reflects the approxmaton qualty of the network 7,9. Such networks can be used to predct non-lnear functon spaces n optmzaton. Several methods to buld a surrogate model for smlar purpose n optmzaton have been tred by Dulkravch et al. 0-. Applcatons lke response surface constructon for mult-objectve optmzaton requre an estmatng functon havng large number of desgn varables. Buldng WNN for such functons s computatonally expensve. So, nstead of preparng a huge lbrary of wavelets 7 an evolutonary based search algorthm s suggested for large dmensonal functon estmaton. In ths paper, we propose a genetc algorthm (GA search for settng the network parameters n a WNN. Several modfcatons to the bass functon and network archtecture are also studed. The networks so created were tested usng many well known test functons to valdate ther accuracy and robustness. Fnally, we present a hybrd wavelet neural network capable of estmatng hghly non-lnear functon spaces wth good accuracy and usng lesser computatonal resources. Graduate student. Student member AIAA. Professor and Charman of the Department. Drector of MAIDROC. Assocate Fellow AIAA.

2 II. Genetc Algorthm based Wavelet Neural Network Wavelets occur n the famly of functons generated from a mother wavelet ψ ( x. Each wavelet n t s defned by dlaton vector a whch controls the scalng and translaton vector t whch controls the poston. Gven a tranng set, the overall response of a WNN can be arthmetcally wrtten as f N p ( x = w + w x t a 0 ψ ( = where N s the number of wavelet nodes n the hdden layer and s the synaptc weght for each hdden node n p the WNN. The dlaton and translaton vectors have sze equals to the number of varables n the estmated functon. Such a network can be used to approxmate any functon. where f ( + ε w y = f x ( s a regresson functon and the error term ε s a zero-mean random varable of dsturbance. One of the well known approaches for constructng WNN 7 requres the generaton of a wavelet lbrary, W. Ths lbrary s composed of dscretely dlated and translated versons of mother wavelet functon ψ ( x. The next step s selectng the best wavelets based on the tranng data from ths lbrary to buld the regresson. Ths approach for buldng WNN becomes computatonally expensve when the estmated functon has a huge number of varables. Ths s due to exponental ncrease of the sze of the wavelet lbrary W wth the dmenson of the estmated functon. Searchng such a huge lbrary one-by-one s computatonally redundant. Therefore, the stochastc approach for searchng the best wavelets for the WNN hdden nodes s proposed. The concept of bnary GA was used to search for the wavelets requred for the hdden nodes n the WNN. The dlaton and translaton factors (bnary representaton for each dmenson of the wavelet were concatenated to form the chromosomes n the GA populaton. A typcal representaton of a wavelet n -D functon estmaton looks lke t a t a Fgure. Bnary strng representaton of a -D wavelet. It has two dlataton factors specfyng the scalng and two translaton factors specfyng the poston of the wavelet n each dmenson. The varables space s normalzed so the translaton factors can vary wthn [-, ] and the dlaton factors were vared wthn [0., 0.8]. The ftness for selectng the wavelet was defned as the norm of the resdue obtaned by dong Multple Lnear Regresson (MLR of the values gven by the wavelet transform of the tranng data vs. the real functon values. The GA was run for a suffcent number of generatons to select a wavelet. Subsequent wavelets were searched by the GA based on the resdue obtaned n former step set as target values. Ths approach was unable to search for proper wavelets when the number of varables n the estmated functon went beyond ten. The chromosome length for such functons was huge and the bnary GA became neffcent. Therefore, a real numbers GA based search was proposed. In real numbers GA, the wavelet s represented as a strng of real number nstead of a bnary strng. The range for searchng for the values of dlaton factors was relaxed to [0.005, 5.00]. Ths gave more flexblty to the GA for searchng approprate wavelets. A typcal example of a wavelet representaton n -D functon estmaton s a t a t a t a t Fgure. Real strng representaton of a -D wavelet. The ftness assgnment was smlar to the prevous method. In addton, Whole Arthmetc Crossover and Floatng Pont Mutaton operators were used. Separate GAs were run serally for fndng the actvaton functon n each node of the WNN archtecture.

3 III. Modfcatons to Bass Functon and Archtecture Tranng the WNN for response surface generaton was done usng a random dataset. The sze of the dataset requred to reach a gven accuracy can be decreased by usng Sobol Quas-Random Sequence nstead of random tranng data. Sobol sequence scatters ponts n the n-dmensonal space unformly. WNN traned on such a dstrbuton learns the functon space unformly. Ths helps n reducng the computatonal effort needed n tranng a WNN to acheve a gven accuracy. Typcally, the mother wavelet used n the WNN s Mexcan Hat wavelet descrbed by the followng functon. x / ( x = π ( x exp ψ ( Gaussan wavelets were used wth ths mother wavelet to construct the WNN. For each node of the WNN, GA searched the best Mexcan Hat wavelet as well as the best Gaussan Wavelet. The one havng a lower norm of resdue after performng MLR was selected and nserted n the WNN archtecture. The overall archtecture of WNN had a mxture of Mexcan Hat and Gaussan wavelets and looked lke: Fgure. Mxed archtecture of WNN usng Mexcan Hat and Gaussan wavelets To mprove the accuracy of predcton further few other parameters were ntroduced wth dlaton and translaton vectors. An exponent parameter, n, whch controlled the sharpness of the bass functon was used. The mappng of the nput varable for the mother wavelet so obtaned can be expressed as n x t x = ( a Now the GA strng had three parameters per each dmenson of the functon. The lmts for the exponent parameter was set wthn [0.9,.] and produced more accurate predctons. The followng fgures show the mpact of the exponent parameter on the shape of the bass functon used n the WNN. Another parameter to control the skewness was ntroduced n the bass functon. Ths parameter, p, controlled the magntude and drecton of skewness of the bass functon. Followng s the mathematcal descrpton of a skewed Gaussan dstrbuton. x P 5 ( (. ( x = / exp * X ψ (5 π The parameter p can take only ntegral values and the magntude of skewness s governed by ts absolute value. The drecton of skewness s governed by the vector X n the governng equaton.

4 Fgure. Shape of Gaussan bass functon wth exponent parameter n = 0.8 and n =.0 Fgure 5. Plan vew of skewed wavelets: (a negatve x-axs (b negatve x-axs and postve y-axs. Plan vews of -D skewed wavelets are shown above. It shows that the wavelet bass functon can also be skewed n multple drectons smultaneously. The value of parameter p was set between [-5, 5] to determne the proper wavelet for the WNN nodes. All these parameters (a, t, n, and p for each dmenson of the wavelet were searched usng real based GA. Fgure 6. Hybrd WNN archtecture usng full tranng dataset.

5 Ths network showed superor performance for a large varety of test problems. The accuracy of hghly nonlnear functon estmaton was stll a problem. To mprove t further the concept of usng multple WNN n a sngle archtecture was proposed. Ths dea can be mplemented n two dfferent ways. Frst, the tranng dataset can be dvded nto a fxed number of exclusve sets and each of these can be used to tran a network separately. Second, multple networks can be created by tranng each of them wth the complete tranng dataset avalable. After generatng all the traned sub-networks the hybrd WNN was created assemblng each one of them to form a sngle network wth proper weghts. After a few tests, the dea of tranng each sub-network wth a complete tranng dataset was concluded to gve better accuracy. Here, the number of sub-networks S, needed n the fnal archtecture was decded n advance. The full dataset was used to tran s-wnns separately and generate sutable sub-networks. Once all the sub-networks are created, the full tranng data s used to obtan the weghts for each of them. Lnear regresson of the output of each of these networks wth the target value s done n order to obtan the weghts. IV. Test Functons and Testng Scheme For testng dfferent versons of WNN prepared here several mathematcal test functons were developed. The dmenson of test functons vared from 7. Ths would help to evaluate the performance of each of them vs. the dmenson of the estmated functon. These test functons were developed usng many elementary functons for studyng the robustness of WNN. The test functons so developed are tabulated below: Dmenson of Functon 7D 6D 5D D D D Table Test functons developed usng elementary functons Functon Used for Testng + x + x sn(x + exp(x cos(x - log( + x 5 (x (x + x + x sn(x + exp(x cos(x - log( + x 5 (x (x + x + x sn(x + exp(x cos(x - log( + x 5 (x + 7 (x + x + x sn(x + exp(x cos(x - log( + x (x + 7 (x + x + x sn(x + exp(x cos(x - log( + x (x + 7 (x + x + x sn(x + exp(x cos(x - log( + x (x + 7 (x WNNs were traned usng 0 Sobol ponts and were tested usng 0 subsequent Sobol ponts. All the networks had about 8 nodes n the hdden layer of the NN. The tested ponts were dvded nto sx groups dependng on the amount of error for assessment. Fnally, for each of these test functons stacked bar plots of percentage of tested ponts fallng n each group was made n order to compare them. Table Groups for dvdng tested data for evaluaton Group Number 5 6 Error Value 0-0% 0-0% 0-50% 50-00% 00-00% >00% The above testng scheme was used to evaluate the performance of several versons of WNN suggested. Later, few well known test functons 5 were used to evaluate the performance of WNN havng dfferent bass functons and hybrd WNN was also tested. Chen et al. gave metrcs to evaluate the performance of predcton. Most of the test problems had typcal features of engneerng problems. Two relatve scales were consdered (number of varables 0 and number of varables =,. Both low-order nonlnear as well as hgh-order nonlnear problems were used. And fnally, one test problem had nosy behavor. All the test problems obtaned were selected from the book by Hock and Schttkowsk 6 whch offers many other problems for testng nonlnear algorthms. Second order polynomal models have k = ( n + ( n + / coeffcents for n desgn varables. Kaufman et al. 7 found that.5k sample ponts for 5-0 varable problems and.5k sample ponts for 0-0 varable problems were necessary to obtan reasonably accurate second order polynomal models. Therefore, for small scale problems (number of varables =,, 0n data ponts were used. For large scale problems (number of varables > 0, k data ponts were used to tran the WNN. For testng each model, Sobol ponts were used. 7 5

6 The performance of any technque to predct functonal space can be measured n varous aspects lke robustness, effcency, transparency, and conceptual smplcty. To assess the accuracy of the WNN three dfferent metrcs were used: R Square, relatve average absolute error, and relatve maxmum absolute error. The equatons for these three metrcs are gven below: R Square: y n ( y yˆ = MSE R = = n varance = ( y y Here, s the observed value, s the predcted value, and ŷ (6 y s the mean of observed value. MSE (Mean Square Error represents the departure of the model from the real smulaton model, and the varance captures the magntude of rregularty n the problem. The larger the value of R Square, the more accurate the model s. Relatve Average Absolute Error (RAAE: RAAE = n = n y STD yˆ (7 Here STD stands for the standard devaton. The smaller the value of RAAE, the more accurate the model s. Ths s usually hghly correlated wth R Square. Relatve Maxmum Absolute Error (RMAE ( y yˆ, y yˆ,..., y y max ˆ n n RMAE = (8 STD Large RMAE ndcated large error n one regon of the desgn space even though the overall accuracy ndcated by R Square and RAAE can be very good. Ths metrc cannot show the overall performance n the desgn space, so t s not as mportant as the other two. Based on the above scheme, dfferent real parameter GA based WNN were tested. The magntude of the metrcs showed the accuracy of the model and the varance of the metrcs values among dfferent problems llustrates the robustness of the model. V. Results of Varous Test Functons The test results for a dfferent verson of WNN developed are shown n ths secton and ts performance s analyzed. Frst, the mpact of usng Sobol sequence nstead of random tranng data on the effcency of WNN was studed. For ths study a -D test functon was desgned and used to evaluate the performance of WNN traned on Sobol sequence vs. WNN traned on random tranng data. The mathematcal expresson descrbng the test functon s: y = x x sn 5x sn x (9 ( ( ( 6 The tranng of WNN usng random data requred 00 ponts. Smlar accuracy on tranng data could be acheved usng only 00 Sobol ponts to tran the second WNN. The requred accuracy of 0% error on tranng data was acheved n 8.7 seconds usng random data ponts where as the WNN traned wth Sobol data ponts took only 8.56 seconds. The followng plots show the real functon space and the estmated functon space usng Sobol ponts for tranng. Studyng the resdue of estmated functon vs. the number of nodes n the WNN we concluded that the WNN traned wth Sobol ponts reached lower resdue faster than the WNN traned wth random ponts. Therefore, tranng the WNN usng Sobol ponts helped to buld models faster and more accurately. 6

7 Fgure 7. (a The real -D functon space (b The estmated functon space usng Sobol traned WNN. Accuracy of bnary GA based WNN and real GA based WNN n predctng functon spaces were studed usng a set of test functons [Table ]. Followng stacked bar plots shows the percentage of testng data ponts havng partcular ranges of error for several test functons. Real GA Based WNN Real GA Based WNN 00% 00% Percentage of Testng Data Ponts n Each Group 90% 80% 70% 60% 50% 0% 0% 0% 0% >00% 00-00% 50-00% 0-50% 0-0% 0-0% Percentage of Testng Data Ponts n Each Group 90% 80% 70% 60% 50% 0% 0% 0% 0% >00% 00-00% 50-00% 0-50% 0-0% 0-0% 0% D D D 5D 6D 7D 0% D D D 5D 6D 7D Dmenson of Test Functon Dmenson of Test Functon Fgure 8. Stacked bar plots of percentage of tested data havng specfc error: (left bnary GA based WNN, (rght real GA based WNN It shows that the percentage of data havng good predcton (0 0 % error decreases as the dmenson of functon ncreases. Real GA based WNN out performs bnary GA based WNN n all of the test functon and s chosen for future model preparaton. The effect of dmenson of the functon on the computatonal expense requred for tranng the WNN was analyzed. It showed that the tme taken to tran a sngle WNN usng a sngle processor vared lnearly wth the dmenson of the functon. CPU Tme Taken to Construct the WNN CPU Tme (Seconds D D D 5D 6D 7D Dmenson of Tested Functon Fgure 9. Computng tme requred by a sngle processor for buldng WNN for varous test functons. 7

8 The accuracy of predctons for the same test functons were mproved further by addng the concept of mxed archtecture along wth the exponent parameter p. The followng stacked bar plot shows the mprovement for each of the test functons. Real GA Based WNN: Addton of Exponent Term Percentage of Testng Data Ponts n Each Group 00% 90% 80% 70% 60% 50% 0% 0% 0% 0% 0% >00% 00-00% 50-00% 0-50% 0-0% 0-0% D - OLD D - NEW D - OLD D - NEW D - OLD D - NEW 5D - OLD 5D - NEW 6D - OLD 6D - NEW 7D -OLD 7D - NEW Dmenson of Test Functon Fgure 0. Improvement n accuracy of WNN usng mxed archtecture and exponent parameter. Ths extra parameter helped n fndng a more accurate descrpton of the estmated functon locally. The GA searches for the proper set of parameters to construct the WNN effcently. The current verson of WNN was further tested usng thrteen test problems provded by Chen et al. 5. The performance metrcs were compared to evaluate the performance of WNN n predctng varous functons. The value of R-Square shows the average accuracy of predcton and was used prmarly for performance evaluaton. R-Square value of one s produced when the predctons are 00% accurate and a value of zero shows random predctons. The followng plot shows the value of R-Square for each of the test functons gven by WNN when prepared usng three dfferent types of bass functons: Gaussan, Mexcan Hat and Skewed Gaussan (Ch-Square. The values of R-Square for most of the test problems usng Gaussan and Mexcan Hat wavelet approaches one. Thus, such bass functons were sutable for buldng accurate models. Few problems (,, 9, and 0 had lower values of R-Square and mpled that the WNN was unable to predct these wth good accuracy. These problems were hghly non-lnear. WNN was able to predct problem number wth good accuracy, though the problem had a lot of nose. To mprove the WNN to predct hghly non lnear functons more accurately, hybrd WNN were used. Two hybrd WNNs were prepared and tested usng the same test problems. One of them had sub-networks traned usng exclusve sets of the whole tranng dataset and the other had sub-networks traned on the full tranng dataset. The results were compared wth the predctons usng sngle Mexcan Hat WNN. Clear mprovements n predctons were seen usng hybrd WNN havng sub-networks traned on full tranng dataset. Hybrd WNN wth sub-networks traned on excusve sets of tranng data dd predct better then sngle WNN (problem number 6 and 0 but had worse predctons (problem number and also. Therefore, hybrd WNN wth sub-networks traned on full tranng dataset was suggested for hghly non-lnear functon spaces. About 5 6 sub-networks are recommended n the archtecture of hybrd WNN. Fnally, the performance of the sngle network WNN vs. the dmenson of estmated functon was studed. Problem number was chosen 5 and the number of varables was vared to get several test functons. WNN was traned on 00 Sobol ponts and the model was confrmed usng 00 testng ponts. The accuracy of predctons and the computatonal expense was compared among these test functons. The followng plot compares the R-Square values and the tme taken by a sngle processor to generate the model for the test functons havng dfferent number of dependent varables. 8

9 Comparson of R Square for varous wavelets used. Gaussan Wavelet Mexcan Hat Wavelet ChSQ Wavelet (Skewed 0.8 R Square PB PB PB PB PB5 PB6 PB7 PB8 PB9 PB0 PB PB PB Problem Number Fgure. R-square values for test functons obtaned usng dfferent bass functons. R Square Comparson of R Square for varous wavelets used Mexcan Hat Wavelet Hybrd WNN(EXCLUSIVE Hybrd WNN(FULL TRAINING 0 PB PB PB PB PB5 PB6 PB7 PB8 PB9 PB0 PB PB PB Problem Number Fgure. R-square values for test functons usng varous hybrd WNN. Fgure shows that the accuracy of the model predcton decreases as the number of desgn varables ncreases. One reason for ths s the constant number of tranng ponts used to buld each WNN. Also, the tme taken by a sngle processor to buld varous WNN models goes on ncreasng wth the dmenson of the estmated functons. The GA takes longer to search for the proper nodes as the dmenson of the functon ncreases. Assumng the value of R Square > 0.5 s good predcton, one can predct functon havng 90 varables effectvely (extrapolatng. Thus, ths algorthm can predct hgh dmensonal functon space, but the level of functon non-lnearty s always a factor governng the accuracy of predcton. 9

10 . RSQ CPU Tme R Square CPU tme # of varables 0.00 Fgure. Varaton of accuracy and computatonal expense wth dmenson of test functon. VI. Conclusons Ths paper presents a novel technque to search the proper actvaton functons for the constructon of WNN. Such a method helps to buld response surface for predctng hgh dmensonal functon spaces accurately and effcently. Several modfcatons to the archtecture and bass functon n WNN were suggested and tested usng dverse test functons. Smulaton results show that WNN for predctng functon havng large number of varables (~90 can be mplemented accurately. The modfed WNN developed n such a way had about 5 7 % average absolute error on tranng data n most of the test problems. The fnal testng for most of the test functons was done usng data ponts. Wth such large testng dataset we can assume that any further predcton of the functonal space wll have smlar accuracy. It was found that about ten to twelve actvaton nodes n the hdden layer of WNN were adequate for good predctons.e. 5% average absolute error on tranng data. Further addton of actvaton nodes n WNN mproved the accuracy on tranng data only slghtly, but dd not help n mprovng the accuracy on testng data further. Ths mples that for the gven dataset a WNN wth about ten to twelve actvaton nodes extracted most of the nformaton regardng the topology of the functonal space. Fnally, a hybrd WNN network was developed usng the concept of havng multple WNNs and helped n ncreasng the accuracy of predcton of hghly non-lnear functonal spaces. Acknowledgement The authors are grateful for the fnancal support provded for ths work by the US Ar Force Offce of Scentfc Research grant FA montored by Dr. Todd E. Combs and by the US Army Research Offce/Materals Drectorate under the contract number DAAD montored by Dr. Wllam M. Mullns. References Chen, D. S. and Jan, R. C., A Robust Back Propagaton Learnng Algorthm for Functon Approxmaton, IEEE Trans. Neural Networks, Vol. 5, May 99, pp Hwang, J.N., Nonparametrc Multvarate Densty Estmaton: A Comparatve Study, IEEE Trans. Sgnal Processng, Vol., 99, pp Shashdhara, H. L., Lohan, S. and Gadre, V. M., Functon Learnng usng Wavelet Neural Networks, Proceedngs of IEEE Internatonal Conference on Industral Technology, Vol., 000, pp Crstea, P., Tuduce, R. and Crstea, A., Tme Seres Predcton wth Wavelet Neural Networks, In Proceedngs of the 5 th semnar on IEEE Neural Network Applcatons n Electrcal Engneerng, 000. pp Ho, D. W. C., Zhang; P.-A. and Xu, J., Fuzzy Wavelet Networks for Functon Learnng, IEEE Transactons on Fuzzy Systems, Vol. 9, No., Feb 00, pp Zhang, J., Walter, G. G. Mao, Y. and Lee, W. N. W., Wavelet Neural Networks for Functon Learnng, IEEE Trans. Sgnal Processng, Vol., June 995, pp Zhang, Q., Usng Wavelet Network n Nonparametrc Estmaton, IEEE Trans. Neural Network, Vol. 8, 997, pp

11 8 L, S. T. and Chen, S. C., Functon Approxmaton usng Robust Wavelet Neural Network", Proceedngs of the th IEEE Internatonal Conference On Tools wth Artfcal Intellgence (ICTAI 00, Washngton D.C., November -6, 00, pp Zang, Q. and Benvenste, A., Wavelet Networks, IEEE Transactons on Neural Networks, Vol., No. 6, November Dulkravch, G. S. and Egorov, I. N., Robust Optmzaton of Concentratons of Alloyng Elements n Steel for Maxmum Temperature, Strength, Tme-To-Rupture and Mnmum Cost and Weght, ECCOMAS Computatonal Methods for Coupled Problems n Scence and Engneerng, Fra, Santorn Island, Greece, May 5-8, 005. Dulkravch, G. S. and Egorov, I. N., Optmzaton of Alloy Chemstry for Maxmum Stress and Tme-to- Rupture at Hgh Temperature, AIAA paper 00-8, 0th AIAA/ISSMO Multdscplnary Analyss and Optmzaton Conference, eds: A. Messac and J. Renaud, Albany, NY, Aug. 0 Sept., 00. Egorov-Yegorov, I. N. and Dulkravch, G. S., Chemcal Composton Desgn of Superalloys for Maxmum Stress, Temperature and Tme-to-Rupture Usng Self-Adaptng Response Surface Optmzaton, Materals and Manufacturng Processes, Vol. 0, No., May 005, pp Wrght, A. H., Genetc Algorthms for Real Parameter Optmzaton. In G. J. E. Rawlns (Ed., Foundatons of Genetc Algorthms, 99, pp Sobol, I. and Levtan, The Producton of Ponts Unformly Dstrbuted n a Multdmensonal Cube, Preprnt IPM Akad. Nauk SSSR, Number 0, Moscow Jn, R, Chen, W. and Smpson, T., Comparatve Studes of Metamodelng Technques under Multple Modelng Crtera, Journal of Structural Optmzaton, (, 00, pp Hock, W. and Schttkowsk, K., 98, Test Examples for Nonlnear Programmng Codes, Lecture Notes n Economcs and Mathematcal Systems, Vol. 87, Berln / Hedelberg / New York: Sprnger-Verlag. 7 Kaufman, M., Balabanov, V., Burgee, S. L., Gunta, A. A., Grossman, B., Mason, W. H. and Watson, L. T., Varable Complexty Response Surface Approxmatons for Wng Structural Weght n HSCT Desgn, Proc. th Aerospace Scences Meetng and Exhbt, Reno, NV, AIAA Paper , 996.